11-1 Goodness-of-Fit 587 3. Cybersecurity The accompanying Statdisk results shown in the margin are obtained from the data given in Exercise 1. What should be concluded when testing the claim that the leading digits have a distribution that fits well with Benford’s law? 4. Cybersecurity What do the results from the preceding exercises suggest about the possibility that the computer has been hacked? (Normal Internet traffic has a distribution that fits well with Benford’s law.) Is there any corrective action that should be taken? In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and, or critical value, and state the conclusion. 5. Heights Measured or Reported? A random sample of the last digits of heights (in.) of males from Data Set 4 “Measured and Reported” is summarized in the table below. Use these last digits to determine whether they occur with about the same frequency. Use a 0.05 significance level. Do the corresponding heights appear to be measured or reported? Last Digit 0 1 2 3 4 5 6 7 8 9 Frequency 12 9 8 7 9 9 12 10 11 14 6. Heights Measured or Reported? Repeat the preceding exercise using the frequencies in the following table, which summarizes all of the 2784 male heights listed in Data Set 4 “Measured and Reported.” Does the larger data set have much of an effect on the results from Exercise 5? Last Digit 0 1 2 3 4 5 6 7 8 9 Frequency 321 315 329 200 202 203 285 303 297 329 7. Testing a Slot Machine The author purchased a slot machine (Bally Model 809) and tested it by playing it 1197 times. There are 10 different categories of outcomes, including no win, win jackpot, win with three bells, and so on. When testing the claim that the observed outcomes agree with the expected frequencies, the author obtained a test statistic of x2 = 8.185. Use a 0.05 significance level to test the claim that the actual outcomes agree with the expected frequencies. Does the slot machine appear to be functioning as expected? 8. Flat Tire and Missed Class A classic story involves four carpooling students who missed a test and gave as an excuse a flat tire. On the makeup test, the instructor asked the students to identify the particular tire that went flat. If they really didn’t have a flat tire, would they be able to identify the same tire? The author asked 41 other students to identify the tire they would select. The results are listed in the following table (except for one student who selected the spare). Use a 0.05 significance level to test the author’s claim that the results fit a uniform distribution. What does the result suggest about the likelihood of four students identifying the same tire when they really didn’t have a flat? Tire Left Front Right Front Left Rear Right Rear Number Selected 11 15 8 6 9. Bias in Clinical Trials? Researchers investigated the issue of race and equality of access to clinical trials. The following table shows the population distribution and the numbers of participants in clinical trials involving lung cancer (based on data from “Participation in Cancer Clinical Trials,” by Murthy, Krumholz, and Gross, Journal of the American Medical Association, Vol. 291, No. 22). Use a 0.01 significance level to test the claim that the distribution of clinical trial participants fits well with the population distribution. Is there a race>ethnic group that appears to be very underrepresented? Race, ethnicity White non-Hispanic Hispanic Black Asian, Pacific Islander American Indian, Alaskan Native Distribution of Population 75.6% 9.1% 10.8% 3.8% 0.7% Number in Lung Cancer Clinical Trials 3855 60 316 54 12
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